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Compute \(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} + \dots + \frac{1}{37 \times 40}.\)

 Apr 29, 2024
 #1
avatar+128772 
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Using  partial fraction decomposition :

1/ [n * (n + 3)] = A / n  + B / (n +3)             multiply through by n *(n + 3)

1 = A (n + 3)  + Bn

1 = (A + B)n  + 3A

3A  = 1

A = 1/3

A+B  = 0

B = -1/3

 

So    

 

1 / [1x 4]  =   (1/3)/1 - (1/3)/4 =   1/3 - 1/12

1/ [ 4 x 7] =   (1/3)/4 - (1/3)/7  =  1/12 - 1/21

1/[7 * 10] =   (1/3)/7 - (1/3)/10 =  1/21 - 1/30

......

1/(34 *37)  = ((1/3)/34 - (1/3)/37  = 1/ 102 - 1/111

1 / (37 * 40) =  (1/3)/(37) - (1/3)/40  =  1/111  - 1/120

 

All the intermediate terms  "cancel"     and  we  are left with

 

1/3  -  1/120   =

 

[ 120 - 3 ] / 360   =

 

117 / 360    =

 

13 /  40

 

cool cool cool

 Apr 30, 2024

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